Article

Optimal Execution Strategies in Limit Order Books with General Shape Functions

Quantitative Finance (Impact Factor: 0.65). 08/2007; 10(2). DOI: 10.2139/ssrn.1510104
Source: arXiv

ABSTRACT

We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (200516.

Obizhaeva , A and
Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009) [CrossRef]View all references) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (200516.

Obizhaeva , A and
Wang , J . 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009) [CrossRef]View all references). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.

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    • "In the literature on mathematical finance, various models of the limit order book have been recently studied, mainly from the point of view of the agents who submit the limit orders. In [11] [13] [7] prices range over a discrete set of values, while in [10] [12] [1] prices are continuous and the shape of the limit order book is described by a density function. An important achievement of these models is that, as soon as the shape of the limit order book is given, this in turn determines a corresponding price impact function, describing how the bid and ask prices change after the execution of a market order. "
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    ABSTRACT: A one-sided limit order book is modeled as a noncooperative game for several players. An external buyer asks for an amount X > 0 of a given asset. This amount will be bought at the lowest available price, as long as the price does not exceed an upper bound . One or more sellers offer various quantities of the asset at different prices, competing to fulfill the incoming order. The size X of the order and the maximum acceptable price are not a priori known, and thus regarded as random variables. In this setting, we prove that a unique Nash equilibrium exists, where each seller optimally prices his assets in order to maximize his own expected profit. Furthermore, a dynamics is introduced, assuming that each player gradually adjusts his pricing strategy in reply to the strategies adopted by all other players. In the case of (i) infinitely many small players or (ii) two large players with one dominating the other, we show that the pricing strategies asymptotically converge to the Nash equilibrium.
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    • "Often the main purpose is volatility estimation, while trading applications are not studied: the complexity of these models leads to high dimensional equations in control problems, difficult to treat both analytically and numerically. (ii) High frequency trading problems: another important literature stream focuses on trading problems in the limit order book: stock liquidation and execution problems ([4], [3], [11], [26], etc ...), or market making problems ([6], [16], [25], [24], [21], [12], etc ...). These papers use stochastic control methods to determine optimal trading strategies, and they are mostly based on classical models for asset price, typically arithmetic or geometric Brownian motion, while the market order flow is usually driven by a Poisson process independent of the continuous price process. "
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    ABSTRACT: We study a an optimal high frequency trading problem within a market microstructure model designed to be a good compromise between accuracy and tractability. The stock price is driven by a Markov Renewal Process (MRP), while market orders arrive in the limit order book via a point process correlated with the stock price itself. In this framework, we can reproduce the adverse selection risk, appearing in two different forms: the usual one due to big market orders impacting the stock price and penalizing the agent, and the weak one due to small market orders and reducing the probability of a profitable execution. We solve the market making problem by stochastic control techniques in this semi-Markov model. In the no risk-aversion case, we provide explicit formula for the optimal controls and characterize the value function as a simple linear PDE. In the general case, we derive the optimal controls and the value function in terms of the previous result, and illustrate how the risk aversion influences the trader strategy and her expected gain. Finally, by using a perturbation method, approximate optimal controls for small risk aversions are explicitly computed in terms of two simple PDE's, reducing drastically the computational cost and enlightening the financial interpretation of the results.
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    • "The traditional point of view to this problem (e.g. (Alfonsi and Schied, 2010), and (Alfonsi et al., 2007) ), optimal liquidation depends on exiting the sufficiently large limit order. As consequence, price impact is function of the shape and depth of the Limit Order Book (LOB). "
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    ABSTRACT: In an illiquid market as a result of a lack of counterparties and uncertainty about asset values, trading of assets is not being secured by the actual value. In this research, we develop an algorithmic trading strategy to deal with the discrete optimal liquidation problem of large order trading with different market microstructures in an illiquid market. In this market, order flow can be viewed as a Point process with stochastic arrival intensity. Interaction between price impact and price dynamics can be modeled as a dynamic optimization problem with price impact as a linear function of the self-exciting dynamic process. We formulate the liquidation problem as a discrete-time Markov Decision Processes where the state process is a Piecewise Deterministic Markov Process (PDMP), which is a member of right continuous Markov Process family. We study the dynamics of a limit order book and its influence on the price dynamics and develop a stochastic model to retain the main statistical characteristics of limit order books in illiquid markets.
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